David P. Sanders

Use the standard (non-interval) Newton method to look for approximate roots, then use epsilon inflation to prove the existence of a root and remove boxes there.

Remove old code

3

Remove old code: - [x] bisection.jl - [x] move out definitions of `N` used in `branch_and_prune` - [ ] krawczyk.jl - [ ] newton.jl - [ ] find_roots stuff

No method roots(f, v::Vector{IntervalBox})

2

I'm sure this used to work.

Failure to find solutions for a 10-dimensional problem

13

Problem taken from https://discourse.julialang.org/t/solvers-fail-on-nonlinear-problem-which-has-solutions/20051: ``` function f(X, p) x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 = X return SVector( -p[17] * x1 + -2 * p[1] * (x1 ^ 2 / 2) + 2 * p[2]...

Newton fails with abs

6

``` julia> f(p) = abs(1 / ( (1+p)^30 ) * 10_000 - 100) f (generic function with 1 method) julia> roots(f, -1000..1000, Newton) 2-element Array{Root{Interval{Float64}},1}: Root(∅, :unique) Root(∅, :unique) ```

Done in #110.

Add centered forms

2

Add mean value forms for scalar and vector functions, and a 3rd-order Taylor form for scalar functions.

Make a RootProblem type

1

It may be useful to have a `RootProblem` type that encodes the function, the `IntervalBox`, the dimension, etc.

Done in #88